On the Dirac delta as initial condition for nonlinear Schrödinger equations

www.elsevier.com/locate/anihpc

On the Dirac delta as initial condition for nonlinear Schrödinger equations

V. Banica

, L. Vega

aDépartement de Mathématiques, Université d’Evry, France bDepartamento de Matematicas, Universidad del Pais Vasco, Spain

Received 14 March 2006; accepted 5 March 2007 Available online 1 September 2007

Abstract

In this article we will study the initial value problem for some Schrödinger equations with Dirac-like initial data and therefore with infiniteL²mass, obtaining positive results for subcritical nonlinearities. In the critical case and in one dimension we prove that after some renormalization the corresponding solution has finite energy. This allows us to conclude a stability result in the defocusing setting. These problems are related to the existence of a singular dynamics for Schrödinger maps through the so-called Hasimoto transformation.

©2007 Elsevier Masson SAS. All rights reserved.

Résumé

Dans cet article on étudie le problème de Cauchy pour des équations de Schrödinger avec donnée initiale de type Dirac et donc avec masseL²infinie, obtenant des résultats positifs pour les non linéarités sous-critiques. Dans le cas critique et en dimension un, on montre qu’après une certaine renormalisation la solution correspondante est d’énergie finie. On en déduit un résultat de stabilité dans le cas défocalisant. Ces problèmes sont liés à l’existence d’une dynamique singulière des applications de type Schrödinger par la transformation de Hasimoto.

©2007 Elsevier Masson SAS. All rights reserved.

MSC:35Q55; 35Q35; 35BXX

Keywords:Schrödinger equations; Singular data

1. Introduction

In this paper we will study the IVP associated to the nonlinear Schrödinger equation (NLS) iut+u± |u|^αu=0, x∈R^d, t >0,

u(0, x)=aδ_x=0+u₀(x) (1)

* Corresponding author.

E-mail addresses:vbanica@univ-evry.fr (V. Banica), mtpvegol@lg.ehu.es (L. Vega).

1 Second author supported by the grant MTM 2004-03029 of MEC (Spain) and FEDER. Both authors supported by the European Project HYKE (HPRN-CT-2002-00282).

0294-1449/$ – see front matter ©2007 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2007.03.007

with 0α, andu₀ regular and possibly small with respect toaδ_x=0. A related problem that we also consider is to take asu(0, x)a perturbation ofae^ix

2 4 .

Let us recall first what is known about the broader question of the minimal regularity needed to assume in the initial condition u(0, x) so that (1) is well posed. Within the Sobolev class H^s withs0 the answer is positive and well understood, at least from the point of view of local wellposedness, and is due to Ginibre and Velo [12], Cazenave and Weissler [3]. The proof follows a Picard iteration scheme based on the so-called Strichartz estimates. It was observed by Kenig, Ponce and Vega in [17], that ifs <0 Picard’s iteration cannot work due to the lack of uniform continuity of the map datum-solution. Then Vargas and Vega propose in [21] a different class of spaces which are built using the Fourier transform of the initial condition. In the particular case of the one-dimensional cubic NLS they are able to consider a larger class thanL². Finally Grünrock in [14] has extended that result, being able to prove local wellposedness ifu(0, x)satisfies that its Fourier transform is in someL^pforp <∞. Therefore he is just missing the delta function in (1) withd=1 andα=2.

Let us recall next the known explicit solutions of (1) ifu₀=0. It was also noticed in [17], that if there is uniqueness of (1) withu₀=0 then the corresponding solution should be invariant under the Galilean transformations. That is to say for anyν∈R^d,

u_ν(t, x):=e^{−it|ν|2+iν·x}u(t, x−2νt )=u(t, x). (2)

This in turn implies thatu:=u_a,±α, with

u_a,±α(t, x)=f_a(t, x)e^{±iAa,α(t )}, (3)

where

fa(t, x)=a e^i|x|

2 4t

(it )^d/2, (4)

and

Aa,α(t )=

⎧⎪

⎪⎨

⎪⎪

⎩

|a|^α

1−α^d₂t^1−αd2 ifα=2 d,

|a|^2d logt ifα=2 d.

(5)

Notice that

limt↓0f_a(t, x)=aδ_x=0 (S), (6)

and as a conclusion the IVP (1) is ill-posed ifα_d² – see Theorem 1.5 in [17] for a precise statement.

A first natural question is ifu_a,±α is a stable solution in the subcritical caseα <²_d. We denotef2= fL². We have the following result.

Theorem 1.1.Letα <²_d. Foru₀∈L², there exists a timet₀=t₀(a,u₀2)and a unique solutionuof(1)such that u−u_a,±α∈L^p

[0, t0), L^q

∩C

[0, t0), L² with(p, q)any admissible pair,

2 p+d

q =d

2, 2p∞, (d, p)=(2,2).

Moreover if0α1thent0can be taken arbitrarily large.

The way to prove this theorem is to write η(t, x)=e^{∓iAα,a(t )}(u−ua,±α).

Using the fact thatf_ais a solution for the linear equation, we are lead to the equation iηt+η±

|η+f_a|^α− |f_a|^α

(η+f_a)=0,

η(0, x)=u0(x). (7)

Then we see that ifα <_d², the term|f_a|^α is locally integrable with respect to the time variable and therefore the usual Picard iteration scheme works. Moreover, if 0α1, we obtain an a priori control of theL²norm, so that we get a global result in this case.

Another natural question in the subcritical case is to consider a more regular perturbation ofu_a,±α. For this purpose, we introduce the conformal transformation

T (f )(t, x)= e^i|x|

2 4t

(it )^d/2f 1 t,x

t

. (8)

Letwbe defined byu=T w. Thenusolves (1) for 0< t < t₀iffwsolves for 1/t0< t <∞

−iwt+w± 1

t^2−αd2|w|^αw=0.

That is iffv(t, x)=w(t, x)e^{∓iAa,α(1/t )}solves for 1/t0< t <∞the equation

−ivt+v± 1 t^2−αd2

|v|^α− |a|^α

v=0. (9)

Let us notice that by the changes of variable we did, u(t, x)=T

e^{±iAa,α(1/·)}v(·,·)

(t, x)=e^{±iAa,α(t )}T (v)(t, x),

the initial solution u_a,±α of (1) corresponds to the constant trivial solutiona of (9). Finally, the equation of the perturbation(t, x)=v(t, x)−a, with initial data at time 1/t0, writes

⎧⎨

⎩

−it+± 1 t^2−αd2

(|+a|^α− |a|^α)(+a)=0, (1/t0, x)=₀(x).

(10) We shall study this equation for large times, in appropriate Sobolev spaces, and under suitable conditions onα. The subcritical conditionα <_d²will be crucial in the proofs, since it gets the integrability at infinity of the time-coefficient in the nonlinearity. The asymptotic behavior ofwill give us informations on Eq. (1), and more precisely on the small time behavior of the perturbations around the solutionua,±α. The fact that we have been able to prove Theorem 1.1 directly on the initial equation (1) is related to the fact that the mixed spaces we are working with are invariant by the conformal transformation.

We have the following result.

Theorem 1.2.Letα <²_d, and lets > ^d₂. For0∈H^s, with norm small with respect to|a|, there exists a timet0=t0(a) such that Eq.(10)has a unique solutionin a small ball ofL^∞((1/t0,∞), H^s). Moreover, the wave operator exists and the equation enjoys the property of asymptotic completeness inH^s.

As a consequence, for u₀ small inΣ^s = {(1+ |x|^s)f ∈L²} with respect to|a|, u₀∈H², there exists a time t₀=t₀(a)such that Eq.(1)admits a solution that writes, for all0< t < t₀,

u(t, x)=u_a,±α+a e^i|x|

2 4t

(it )^d/2e^{±iAa,α(t )} x t,1

t

, (11)

for a uniquesmall inL^∞((1/t0,∞), H^s),₀∈L²(x⁴dx).

The proof of the results onis again standard and relies on the fact that in the cases >^d₂,H^sis an algebra included inL^∞. As usual there is the difficulty of the lack of regularity of the nonlinear function appearing in (10), but this is overcome assuming smallness ofwith respect toa. The passage back to the initial equation (1) is done by using the scattering results combined with the asymptotic behavior of the linear Schrödinger evolution.

A case of rougher perturbations of aδ₀ was recently studied by Kita. In [18] he described the structure of the solutions yielded by initial data exactly a sum of two or three Dirac masses.

Finally, let us say a few words about which is the situation in the parabolic setting of the IVP analogous to (1), that is to say

u_t−u± |u|^αu=0, x∈R^d, t >0,

u(0, x)=aδ_x=0. (12)

This equation has been intensively studied. It particular it has been shown by Weissler [22] that in the focusing case (sign−), for α ²_d, there is no uniqueness. For the defocusing case, if only positive solutions are considered, for α < _d² there is a unique solution, and for α ²_d there is no such solution. This result was proved by Brézis and Friedman in [2]. So, for the heat equation, an important difference is made between the focusing and the defocusing case, even forα <²_d. This is not the picture for the Schrödinger equation as we see from the statements of Theorem 1.1 and Theorem 1.2.

Let us consider next the critical case α=_d² and just in dimension d=1. As it is well known this equation is completely integrable. It is also closely related to the Schrödinger map equation

γ_t=γ∧_±γ_xx,

γ (0, x)=γ₀ (13)

withb∧_±c:=A_±(b∧c), and A_±=

1 0 0

0 1 0

0 0 ±1

. Then it is straightforward that

∂_tA±γ , γ =0 (14)

ifγ solves (13). Therefore forA+we get the Schödinger map onto the unit sphereS², and forA−we have it onto 2d hyperbolic space:

H²=

(a₁, a₂, a₃)∈R³such thata²₁+a²₂−a²₃= −1, a3>0 . Equation (13) can be also obtained from the flows of curves inR³given by

χ_t=χ_x∧_±χ_xx. (15)

In the Euclidean case this equation is also called the Local Induction Approximation and was obtained by Da Rios [5] as a crude model which describes the dynamics of a vortex filament equation within Euler equations. In [16], Gutiérrez, Rivas and Vega obtain solutions of the IVP

⎧⎨

⎩

χ_t=χ_x∧₊χ_xx, χ (0, x)=

A⁺₁x, x0, A⁺₂x, x0,

(16) for any unit vectorsA⁺₁, A⁺₂ such that

A⁺₁ +A⁺₂ =0.

More recently de la Hoz [6] has proved a similar result in the nonelliptic setting. Namely he obtains solutions of

⎧⎨

⎩

χt=χx∧−χxx, χ (0, x)=

A⁻₁x, x0, A⁻₂x, x0,

(17) for any pair of vectorsA⁻₁ andA⁻₂ inH².

In both cases the corresponding solution curves are described geometrically by the curvature and the (generalized) torsion given by

c(t, x)= c₀

√t, τ (t, x)= x

2t, (18)

withc₀a free parameter uniquely determined by(A^±₁, A^±₂). Therefore at timet=1 the curves are real analytic while att=0 a corner is developed ifc₀=0. Recall that the flows given in (15) are both reversible in time so that the above solutions are examples of the formation of a singularity in finite time. From (18) we conclude that for all time

∞

−∞

c²(t, x) dx= ∞.

However we shall see below that after some renormalization these curves have finite energy.

The connection of (15) with the 1d cubic NLS was proved by Hasimoto who used the transformation Ψ (t, x)=c(t, x)exp

i

x 0

τ (t, x) dx

. (19)

Although he worked just in the Euclidean case (i.e.∧+in (15)), a similar argument can be given for the other case, see for example [19] and [7]. The final conclusion is that ifχsolves (15) andΨ is defined as in (19) then

iΨt+Ψxx±1 2

|Ψ|²+a(t )

Ψ =0, (20)

for some real functiona(t ). So in the particular case given in (18) we have Ψ (t, x)=c0

e^ix

2

√4t

t =√

ifc₀(t, x).

Then chosena(t )= −|c₀|²/twe get a solution of (20) with Ψ (0, x)=√

ic0δ.

Notice that the coefficient ¹₂which appears in (20) is harmless and can be easily absorbed by the change of variable (t, x)=

2t,√ 2x

.

Hence we are interested in solving onR, around the particular solutionfa, the equation

⎧⎨

⎩iut+u_xx± |u|²−|a|² t

u=0, u(0, x)=aδ_x=0+u₀(x).

(21) Unfortunately we are not able to deal with (21) directly, so that we propose for anyt₀>0 the related problem

⎧⎪

⎪⎨

⎪⎪

⎩

iut+uxx± |u|²−^|a_t^|2

u=0, u(t₀, x)=a^e

ix2 4t0

√it₀ +u₁(x),

(22)

and we look for a backward solution. That is to say for

0< t < t₀. (23)

It is natural to consider as before the conformal transformation u(t, x)=T v(t, x)=e^ix

2

√4t

itv 1 t,x

t

. (24)

Thenusolves (22) for 0< t < t0iffvsolves for 1/t0< t <∞

−ivt+vxx±1 t

|v|²− |a|² v=0, v(1/t0, x)=a+₀(x),

(25) where0is defined byu1(x)=(T 0)(t0, x). It is easy to solve (25) locally in time for both situations focusing and defocussing. Moreover there is a natural energy. In fact if we define

E(t )=1

2 v_x(t, x)²dx∓ 1

4t v(t, x)²− |a|²2

dx, (26)

then ifvis a solution of (25), we have

∂tE(t )∓ 1

4t² v(t, x)²− |a|²2

dx=0. (27)

As a consequence, and in the defocussing situation, we will be able to prove the following theorem which is the main result of this paper.

Theorem 1.3.For allt₀>0and for all₀∈H¹, there exists a unique solution of the IVP

−ivt+vxx−1 t

|v|²− |a|²

v=0, 1/t0< t <∞, v(1/t0, x)=a+₀(x),

(28) with

v−a∈C

(1/t0,∞), H¹ . Moreover

vx(t, x)²dx2E(1/t0), (29)

∞

1 t0

v(t, x)²− |a|²2

dxdt

t² <4E(1/t0), (30)

and in particular lim inf

t→∞

1

t v(t, x)²− |a|²2

dx=0. (31)

Remark 1.4.Using (31) and (24) we obtain that the solutionuof (22) in the defocussing setting for 0< t < t₀

satisfies lim inf

t→0

tu(t )²− |a|²

2=0, (32)

which can be understood as a weak stability result of the singular solutionae^ix

2 4t /√

t of (22).

Remark 1.5.It would be very interesting to be able to compute the limit ofu(t, x)att=0,at least if small perturba- tions ofa^ei

x2

√4t

t are considered. The difficulty is that the time dependent potential ^|a_t^|2, is of long range, and therefore the dynamics at infinity of the renormalized variablev should be modified accordingly; see [20]. This remark quite likely applies for small perturbations even in the focusing situation. We plan to look at these questions somewhere else.

Remark 1.6.It is interesting to write which is the energy for (19), the solution of (20), in terms of the geometric quantitiescandτ. It is given by

E(t ) = t² 4√ 2

+∞

−∞

c²_x(t, x)+c²(t, x) x

2t −τ (t, x) 2

dx+ 1 16√

2 +∞

−∞

t c²(t, x)−c²₀2

dx. (33)

Then d

dtE(t ) − 1 16√

2t +∞

−∞

t c²(t, x)−c²₀2

dx=0, (34)

and

lim inf

t→0

t|c|²− |c₀|²

2=0.

Recall that in this case we will solve the equation backwards in time.

Finally, let us make a remark on the Gross–Pitaevskii defocusing equation iψt+ψ−

|ψ|²−1 ψ=0,

ψ (0, x)=ψ0(x). (35)

This equation was globally solved in 1+H¹(R^d), with an exponential growth in time control of the massψ (t )−12, ford∈ {2,3}by Bethuel and Saut, and ford=1 by Gallo [1,9].

In a class of larger spaces, the Zhidkov spacesX^k(R^d), it has been solved inX¹(R)and inX²(R²)by Zhidkov, by Gallo and by Goubet [23,24,8,13]. Also, considered in the natural energy space{f ∈H_loc¹ ,∇f∈L²,|f|²−1∈L²}, it has been solved ford ∈ {2,3}and ford =4 with smallness assumption, by Gérard in [10]. Recently, Gustafson, Nakanishi and Tsai described in [15] the scattering in modified 1+H^d2−1(R^d)spaces, ford4 and small data. The proof relies on the linearized equation and the conservation of the energy is not used.

By exploiting more the mass and energy laws, as done for proving Theorem 1.3, we can get a slightly modified proof of the very short and simple one in [1], allowing to have the following result.

Proposition 1.7. The solution of (35)is globally well-posed in1+H¹(R^d), for all dimensions d such that local existence occurs, that is surely ford∈ {1,2,3}, with the control

ψ (t )−1

2ct,

where the constantcdepends on the initial data.

The article is structured as follows. The first two sections contain the proofs for the results in the subcritical case.

In Section 4 we prove Theorem 1.3. Section 5 concerns Proposition 1.7, and in the last section we give a technical lemma.

2. The sub-critical power. The Strichartz case

We shall prove the following lemma, that implies the local existence result in Theorem 1.1, as indicated in the introduction.

Lemma 2.1.There existst₀=t₀(a,u₀2)and a unique solutionηof Eq.(7) iηt+η±

|η+f_a|^α− |f_a|^α

(η+f_a)=0, η(0, x)=u₀(x),

such that η∈L^∞

[0, t0), L²

∩L^p

[0, t0), L^q , with(p, q)any admissible couple

2 p+d

q =d

2, 2p∞, (d, p)=(2,2).

Proof. Let us denoteX the intersection of the mixed spaces. In order to do a fixed point argument in a closed ball ofX, we have to estimate the norm of the operator

Φ(η)(t, x)=e^it u₀(x)±i t 0

e^{i(t−τ )}η(τ, x)+f_a(τ, x)^α−f_a(τ, x)^α

η(τ, x)+f_a(τ, x) dτ.

The Schrödinger operator is unitary onL², so one gets Φ(η)(t )

2u02+ t 0

η(τ )+fa(τ )^α−fa(τ )^α

η(τ )+fa(τ )

2dτ. (36)

The nonlinearity we are working with, F (z)=z+f_a(τ, x)^α−f_a(τ, x)^α

z+f_a(τ, x) , verifiesF (0)=0 and

F(z)=max∂_zF (z),∂_zF (z)cf_a(τ, x)^α+z+fa(τ, x)^α . Under the assumptionα0, we get

F(z)cf_a(τ )^α+ |z|^α .

This is the classical growth hypothesis on nonlinearities for proving the local wellposedness. As done for the lo- cal L² Cauchy problem, one can split the nonlinearity in two parts, F =F₁+F₂, with |F₁(z)|c|f_a(τ )|^α and

|F₂(z)|c|z|^α, use

F (z1)−F (z2)= 1 0

F

t z1+(1−t )z2

dt (z1−z2),

for estimatingF (η(τ, x))−F (0)=F (η(τ, x)), and obtain Φ(η)(t )

2u₀2+C t 0

f_a(τ )^αη(τ )

2+η^α+1(τ )

2dτ.

Therefore, for allt >0, Φ(η)(t )

2u02+c(a)ηX

t 0

dτ τ^αd2 +

t 0

η(τ )^α+1

L^2(α+1)dτ.

Now, we compute the second integral, and in the last term we perform a Hölder inequality Φ(η)(t )

2u₀2+c(a)ηXt^1−αd2 +cη^α_L^+p₍¹_[0,t],L^q₎t^4−dα4 , where

p=4(α+1)

dα , q=2(α+1), form an admissible couple. We get

sup

0<tt₀

Φ(η)(t )

2u₀2+c(a)t^1−α

d 2

0 ηX+ct^1−α

d 4

0 η^α_X⁺¹. Then, foru02finite,Φ(η)satisfies the first condition to be inX.

Let us treat now the L^pL^q norm. By using the inhomogeneous global Strichartz inequalities, we obtain that for t >0,

Φ(η)(t )

L^pL^q u02+

t0

0

η(τ )+fa(τ )^α−fa(τ )^α

η(τ )+fa(τ )

2dτ.

The right-hand side term can be treated exactly like the one in (36). In conclusion, ifu₀is inL²,Φ(η)stays inX.

Arguing as before, and assumingt₀small with respect tou₀2and to|a|, we get that the operatorΦ is a contraction on a closed ball ofX. Therefore the lemma follows from the fixed point theorem. 2

Let us prove now the global existence result of Theorem 1.1, in the caseα∈ [0,1]. By multiplying Eq. (7) byη, and by taking then the imaginary part, we have

∂_tη(t )²

2= ∓2 η(t, x)+f_a(t, x)^α−f_a(t, x)^α

f_a(t, x)η(t, x) dx.¯ (37)

Lemma 2.2.Ifα∈ [0,1]there existsc(a) >0such that for allttwe have the a priori estimate η(t )²

2η(t)²

2e^{c(a)t1−α d2}.

Proof. Lemma 6.1 allows us to upper-bound the right-hand side of (37) and get

∂_tη(t )²

2c(a) t^αd2

η(t )²

2, and the lemma follows. 2

Therefore, ifα∈ [0,1], by using Lemma 2.2 we can extend the solutionηfor arbitrary large t0, as done in the critical case in Section 4.2.

3. The sub-critical power. TheH^s case 3.1. Existence of solutions

Lemma 3.1.Lets >^d₂, and let₀∈H^s with 0H^s |a|

8 .

There exists a timet₀=t₀(a)such that Eq.(10), −it+± ¹

t^{2−α d2}

|+a|^α− |a|^α

(+a)=0, (1/t0, x)=₀(x),

has a unique solutionin Y=

f ∈L^∞

(1/t0,∞), H^s , sup

t1/t0

f (t )

H^s |a| 4

.

Proof. In order to do a fixed point argument in this space, we have to estimate the norm inY of the operator Φ()(x, t )=e^−it ₀(x)±i

t 1/t0

e^{−i(t−τ )}(x, τ )+a^α− |a|^α

(x, τ )+a dτ τ^2−αd2

.

By using the fact that the Schrödinger operator is unitary onH^s, Φ()(t )

H^s₀H^s + t 1/t0

(τ )+a^α− |a|^α

(τ )+a

H^s

dτ τ^2−αd2

.

The nonlinearity F (z)˜ =

|z+a|^α− |a|^α (z+a)

is aC^∞function on|z|<^|a₄^| withF (0)˜ =0. Here we see the difference with the case of classical power-nonlinearity

|z|^αz, whose lack of regularity imposes the conditionss < αorαeven. Since(τ )is inY andH^sis embedded inL^∞, it follows that|(τ )|(τ )H^s<^|a₄^|. Therefore [4]

F˜(τ )

H^s c(a), and

Φ()(t )

H^s0H^s +c(a) t 1/t₀

dτ τ^2−αd2

.

By computing the integral, Φ()(t )

H^s ₀H^s+c(a)|t^−1+αd2 −t^1−α

d 2

0 |

|−1+α^d₂| . We are in the caseα <²_d, so

Φ()Y ₀H^s+c(a)t^1−α

d 2

0 .

The smallness of₀inH^sinY yield Φ()Y |a|

8 +c(a)t^1−α

d 2

0 .

Then, fort₀=t₀(a)small enough,Φ()remains inY. Moreover, by arguing similarly, we obtain that the operator acts as a contraction onY. Therefore the fixed point theorem ends the proof. 2

3.2. Scattering properties

Lemma 3.2.Letsan integer such thats >^d₂. (i) For all₊∈H^s with

₊H^s |a| 8 ,

there exists a solutionof (10)inY such that

t→+∞lim (t )−e^−it ₊

H^s=0.

(ii) Let₀∈H^s with ₀H^s |a|

8 ,

and letthe solution given by Lemma3.1. Then there exists a₊small inH^s such that

t→+∞lim (t )−e^−it ₊

H^s=0.

The wave operator of (i) is obtained by doing the same calculus as in the previous subsection, for a fixed point in Y with the operator

Φ()(x, t )=e^−it ₊(x)±i +∞

t

e^{−i(t−τ )}(x, τ )+a^α− |a|^α

(x, τ )+a dτ τ^2−αd2

.

Since the Schrödinger operator is unitary onH^s, proving (ii) is equivalent to proving that e^it (t )

has a limit inH^sastgoes to infinity, and also that

t₁,t₂lim→+∞e^it1(t1)−e^it2(t2)

H^s=0.

By using the Duhamel formulation, it is enough to show

t1,t2lim→+∞

t₂

t₁

(τ )+a^α− |a|^α

(τ )+a

H^s

dτ τ^2−αd2 =0.

The last assertion follows from calculus similar to the one in the proof of Lemma 3.1.

3.3. Proof of Theorem 1.2

Foru₀ small inΣ = {(1+ |x|^s)f ∈L²}, let us define ₊ by the Fourier relation, ˆ₊(x/2)=u₀(x). Then ₊ is small in H^s and Lemma 3.2 ensures us that there exists a timet₀=t₀(a) and a unique solution of (10) in L^∞((1/t0,∞), H^s)behaving at infinity as the free evolution of₊.

Therefore we have the existence of a solution of the equation i∂tu+u± |u|^αu=0,

for all 0< t < t₀, given by u(x, t )=ua,±α+e^i|x|

2 4t

(it )^d2

e^{±iAa,α(t )} 1 t,x

t

. (38)

Let us see now what happens withuat time 0. On one hand, e^i|x|

2 4t

(it )^d2

e^{±iAa,α(t )} 1 t,x

t

−e^−it₊ x t

2

1

t^d2 1

t,x t

−e^−it₊ x t

2

= 1

t

−e^−ti₊ 2

. Then the decay of Lemma 3.2 allows us to say that

limt↓0

e^i|x|

2 4t

(it )^d2

e^{±iAa,α(t )} 1 t,x

t

−e^−ti₊ x t

2

=0. (39)

On the other hand, using the free Schrödinger evolution, e^i|x|

2 4t

(it )^d2

e^{±iAa,α(t )}e^−ti₊ x t

=e^i|x|

2 4t

(it )^d2

e^{±iAa,α(t )}c e^−i|x|

2 4t

(−i/t )^d2

e^−iy24te^ixy2 ₊(y) dy.

So, taking in account limt↓0A_a,α(t )=0,u₀∈H², we have limt↓0

e^i|x|

2 4t

(it )^d2

e^{±iAa,α(t )}e^−ti₊ x t

− ˆ₊ x 2

2

=0. (40)

Therefore, in view of (38), (39) and (40), we conclude thatuverifies the initial condition u(0, x)=aδ_x=0+u₀(x),

so thatuis a solution of Eq. (1).

4. The critical power, defocusing case

In order to treat the defocusing equation (28), for 1/t0< t <∞,

−ivt+v_xx−1 t

|v|²− |a|² v=0, v(1/t0, x)=a+₀(x),

we shall consider instead the equation on=v−a,

−it+_xx−1 t

|+a|²− |a|²

(+a)=0, (1/t0, x)=0(x).

(41)

4.1. A priori estimates

By multiplying Eq. (41) by∂_t, and then by taking its real part, we get

∂_t1

2 _x(t, x)²dx+1

t (t, x)+a²− |a|²

(t, x)+a

∂_t(t, x) dx=0.

Therefore we obtain a nice conservation law, that is,

∂_tE(t )+ 1

4t² (t, x)+a²− |a|²2

dx=0, where

E(t )=1

2 x(t, x)²dx+ 1

4t (t, x)+a²− |a|²2

dx.

By integrating fromttotthe energy law, we obtain E(t )+

t t

(τ, x)+a²− |a|²2

dxdτ

4τ²=E(t).

It follows in particular that for alltt

_x(t, x)²dx2E(t), (42)

and that

(t, x)+a²− |a|²2

dx4t E(t). (43)

Finally, we shall get a control in time of the mass of. By multiplying Eq. (41) byand taking the imaginary part we obtain the mass law

∂_t1

2 (t, x)²dx= −1

t (t, x)+a²− |a|²

a(t, x) dx.¯ (44)

By performing a Cauchy–Schwarz inequality in space, we obtain

∂t(t )²

22|a| t (t )

2(t )+a²− |a|²

2. Now we use the upper-bound (43) and get for alltt

∂_t(t )²

24|a|√ E(t)

√t

(t )

2.

By integrating fromttotit follows that for alltt, (t )2(t)

2+c

a, (t)√ t−√

t

c

a, t, (t)√

t , (45)

where the constant depends ona, ont, on the energy andL²norms of(t).

4.2. Global existence inH¹

First, we solve Eq. (41) locally in time. We perform the classical argument of fixed point, done for the cubic equation with constant coefficients (see for example [11], first part of Proposition 4.2). In the proof, every time that the time t⁻¹ appears in the nonlinear terms, we upper-bound it byt₀. Therefore we can construct inX, the space introduced in Section 2, a solution of (41) living on(1/t0, T ), whereT has to verify

c(a) T+T¹² 202

2

1

2.

So it is sufficient thatT verifies

T c(a)

1+ ₀⁴₂.

In the constant coefficient case, this is enough to infer global existence inL², since the mass is conserved. In our case, we shall use the control (45).

Suppose the maximum time of existence is a certain finiteT. We shall prove that the solution can be defined also afterT, and therefore the global existence is implied. Letδ a small positive number, to be chosen later. By treating the problem with initial value at timeT −δ, we can extend the solutionfor all timehsatisfying

h c(a)

1+ (T −δ)⁴₂.

By using the control (45), it follows that we can choose such ahverifying furthermore h c(a, t₀, ₀)

1+(T−δ)²₀⁴₂.

Now, we can chooseδsmall enough, such that c(a, t₀, ₀)

1+(T−δ)²0⁴₂ > c(a, t₀, ₀) 1+T²0⁴₂ > δ,

so it follows thath > δand so we have extendedafter the timeT.

Therefore we have global existence inL², provided that the initial value₀ is finite in energy andL²norms. By using the Gagliardo–Nirenberg inequalities in the energy definition, it is then enough to have0inH¹. Finally, in view of the observation (42), the gradient of the solution remains bounded in time, so the global existence is valid inH¹.

5. A remark on the global existence for the Gross–Pitaevskii equation By denoting in (35)u=ψ−1, we have

i∂tu+u−

|u+1|²−1

(u+1)=0, u(0, x)=ψ₀(x)−1.

We multiply the equation with∂tu, integrate in space and take the real part. We get that the energy E(t )=1

2 ∇u(t, x)²dx+1

4 u(t, x)+1²−12

dx,

is conserved in time. Next, we multiply the equation byu, integrate in space and take the imaginary part. It follows that

1

2∂tu(t )²

2= u(t, x)+1²−1

u(t, x)+1

¯

u(t, x) dx= u(t, x)+1²−1

¯

u(t, x) dx.

By performing a Cauchy–Schwarz inequality in space we get

∂_tu(t )²

22u(t )+1²−1

2u(t )

2, and so

∂tu(t )

2u(t )+1²−1

2.

The energy conservation allows us to conclude that

∂_tu(t )

22

E(0).

Therefore we have obtained the claimed a priori bound on the mass u(t )

22

E(0) t+ u₀2,

which allows the passage to global existence.

6. A technical lemma

Lemma 6.1.Letx,y be complex numbers, andr0. Then |x+y|^r− |y|^rc

|y|^r−1|x| + |x|^r . Moreover, if0r1then

|x+y|^r− |y|^rc|y|^r−1|x|.

Finally, ifxis small with respect toy, then the last estimate is true for allr0.

Proof. By changingx=zy, we have to show forr0, |z+1|^r−1c

|z| + |z|^r ,

and for 0r1, or forzsmall with respect to 1, |z+1|^r−1c|z|.

If|z|¹₂ then

|z+1||z| +13|z|, and sincer0,

|z+1|^r3^r|z|^r. In conclusion,

|z+1|^r−13^r|z|^rc

|z| + |z|^r .

If|z|¹₂ then 0 is not in the intervalI= [min{1,|z+1|},max{1,|z+1|}]. We consider the functionf (x)=x^r defined onI and we get by the mean value theorem

|z+1|^r−1|z+1| −1r sup

α∈{1,|z+1|}

α^r−1|z|r

1+ |z+1|^r−1

. (46)

Ifr1, then

|z+1|^r−1

|z| +1r−1

2^r−1, and we get from (46) that

|z+1|^r−1c|z|. Ifr <1, then

|z+1|1− |z|1 2, and

|z+1|^r−12^−(r−1), and it follows again from (46) that

|z+1|^r−1c|z|.

So, under smallness conditions on|z|with respect to 1, forr0, we have |z+1|^r−1c|z|.

Otherwise, for generalz, we get only |z+1|^r−1c

|z| + |z|^r .